Match ends, Chelsea 2, Everton 1.
Everton caretaker manager David Unsworth said his team did him proud despite their Carabao Cup last-16 defeat by Chelsea.
Willian scored the decisive second as Chelsea advanced to the quarter-finals but they were pushed to the final whistle by the rejuvenated visitors.
Chelsea had contained Everton with ease in the first half and led through Antonio Rudiger's looping header.
But the Toffees were sharper after the break as Kevin Mirallas had a shot well saved and Ademola Lookman hit the bar.
Willian drilled a superb low finish home in stoppage time before Dominic Calvert-Lewin netted a consolation.
"I'm a bad loser but I have to say I'm proud of the performance, they were terrific," Unsworth, who has replaced the sacked Ronald Koeman for now, told BBC Radio 5 live.
"It was a committed performance. Hopefully we'll show that in every game I am in charge."
Chelsea will learn their last-eight opponents when the draw is made at 16:00 BST on Thursday.
Unsworth's first audition shows promise
Unsworth - taking charge of Everton in the wake of Koeman's sacking on Monday - will not have a last-eight tie to look forward to but was rightly delighted with his team's performance.
A commanding defender during his playing days, he will not relish reviewing the opener however.
With Michael Keane dropped to the bench and a flat back four on the pitch, neither Phil Jagielka nor Ashley Williams commanded the penalty box as Rudiger floated free to direct a fine header back across Jordan Pickford from Charly Musonda's cross.
Unsworth has made no secret of his desire to be permanent Toffees manager and, after an insipid first half, his contagious passion for the club appeared to inspire a much-improved second-half showing.
Everton's energy and industry flustered Rudiger, Andreas Christensen and Gary Cahill in the home defence and it took a string of smart saves from Willy Caballero to preserve their lead before Willian drilled in the hosts' second.
A more mobile attack, with Tom Davies, Mirallas and Aaron Lennon all brought into the starting line-up, provided the Toffees with more direction and urgency and that may be the key lesson from the evening.
Conte makes do and wins
With a Champions League campaign to fight alongside their Premier League title defence, this was the type of game to test a squad that Chelsea manager Antonio Conte feels is stretched too thin.
However Danny Drinkwater, a £35m signing from Leicester, is some impressive back-up to call upon and the England international delivered a typically assured performance on his debut for the club.
Michy Batshuayi could not take his chance to similarly press his claims for a first-team role.
The Belgian was outmuscled by Jagielka after rounding Pickford in his only clear chance and saw Alvaro Morata, firmly Conte's first-choice striker, come on in his place before the end.
The moment of quality that killed the game as a contest came from another Conte favourite as Willian hammered a low shot in off the inside of the post to double the lead.
Calvert-Lewin bundled home a minute later, but it was destined to be in vain.
Man of the match - Willian
Ampadu becomes Chelsea's first millennial player - stats
- Chelsea have reached the League Cup quarter-final for the 21st time in their history - only Arsenal (26), Spurs (23) and Liverpool (22) have achieved it on more occasions.
- Chelsea have not lost a home game against Everton in all competitions since November 1994 - remaining unbeaten in their last 28, winning 17, drawing 11.
- Everton have only kept one clean sheet in their last 14 games in all competitions (against Sunderland in the League Cup third round).
- Willian's goal was his first of the season for Chelsea, and ended a run of 14 games in all competitions without scoring (since May v Sunderland).
- Ethan Ampadu (17 years and 40 days old) became the first player born in 2000 to start for Chelsea in any competition.
- Chelsea scored with two of their three shots on target in this game, while Willian's goal was their only one of the second half.
Chelsea travel to Bournemouth in Saturday's tea-time kick-off in the Premier League while Everton visit Leicester on Sunday.
- 2RüdigerBooked at 66mins
- 6DrinkwaterSubstituted forFàbregasat 62'minutes
- 23BatshuayiSubstituted forMorataat 85'minutes
- 17MusondaSubstituted forPedroat 70'minutes
- 37Dos Reis Carvalho
- 6JagielkaBooked at 61mins
- 5WilliamsBooked at 28mins
- 12LennonSubstituted forLookmanat 73'minutes
- 16McCarthyBooked at 52minsSubstituted forCalvert-Lewinat 64'minutes
- 26DaviesBooked at 50mins
- 10RooneySubstituted forNiasseat 81'minutes
- 18G Sigurdsson
- Neil Swarbrick
- Shots on Target
Second Half ends, Chelsea 2, Everton 1.
Goal! Chelsea 2, Everton 1. Dominic Calvert-Lewin (Everton) left footed shot from the centre of the box to the centre of the goal.
Goal! Chelsea 2, Everton 0. Willian (Chelsea) right footed shot from the left side of the box to the bottom right corner. Assisted by Cesc Fàbregas following a corner.
Corner, Chelsea. Conceded by Jonjoe Kenny.
Pedro (Chelsea) wins a free kick in the defensive half.
Foul by Tom Davies (Everton).
Attempt missed. Ademola Lookman (Everton) right footed shot from outside the box misses to the left. Assisted by Leighton Baines following a set piece situation.
Foul by Davide Zappacosta (Chelsea).
Leighton Baines (Everton) wins a free kick on the left wing.
Offside, Chelsea. Cesc Fàbregas tries a through ball, but Álvaro Morata is caught offside.
Attempt blocked. Davide Zappacosta (Chelsea) left footed shot from the right side of the box is blocked. Assisted by Álvaro Morata.
Substitution, Chelsea. Álvaro Morata replaces Michy Batshuayi.
Cesc Fàbregas (Chelsea) wins a free kick in the defensive half.
Foul by Oumar Niasse (Everton).
Substitution, Everton. Oumar Niasse replaces Wayne Rooney.
Ademola Lookman (Everton) hits the bar with a left footed shot from outside the box. Assisted by Kevin Mirallas.
Attempt missed. Michy Batshuayi (Chelsea) left footed shot from a difficult angle and long range on the right is high and wide to the left. Assisted by Davide Zappacosta.
Delay over. They are ready to continue.
Delay in match Michy Batshuayi (Chelsea) because of an injury.